Math Problem Solver

$x^{72} + x^{66} + x^{54} + x^{24} + x^{6} + 1 \in \left\{1 + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}, 1 + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{72} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{66}, 1 + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}, 1 + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{72} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{54} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{24} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{6} + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{66}\right\}$

$i$ is the imaginary unit, defined as $i^2 = -1$.

By substituting this value into the expression, we find that: $$\begin{aligned}x^{72} + x^{66} + x^{54} + x^{24} + x^{6} + 1 &= 1 + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}\\&= 1 + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{66} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{54} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{24} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{6} + \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{72}\end{aligned}$$